The stable Morse number as a lower bound for the number of Reeb chords

Abstract

Assume that we are given a closed chord-generic Legendrian submanifold $\Lambda \subset P\times \mathbb{R}$ of the contactisation of a Liouville manifold, where $\Lambda$ moreover admits an exact Lagrangian filling $L_{\Lambda }\subset \mathbb{R}\times P\times \mathbb{R}$ inside the symplectisation. Under the further assumptions that this filling is spin and has vanishing Maslov class, we prove that the number of Reeb chords on $\Lambda$ is bounded from below by the stable Morse number of $L_{\Lambda }$. Given a general exact Lagrangian filling $L_{\Lambda }$, we show that the number of Reeb chords is bounded from below by a quantity depending on the homotopy type of $L_{\Lambda }$, following Ono-Pajitnov's implementation in Floer homology of invariants due to Sharko. This improves previously known bounds in terms of the Betti numbers of either $\Lambda$ or $L_{\Lambda }$.

37 pages, 2 figures, final version

References in corpus (3)

• Loose Legendrian embeddings in high dimensional contact manifolds
• Lifting pseudo-holomorphic polygons to the symplectisation of $P\times \mathbb{R}$ and applications
• The lower central and derived series of the braid groups of the sphere and the punctured sphere

Cited by in corpus (1)

• The number of Hamiltonian fixed points on symplectically aspherical manifolds